Theoretical and computational study of several linearisation
techniques for Binary Quadratic Problems
Fabio Furini
1and Emiliano Traversi
21PSL, Universit´e Paris Dauphine, CNRS, LAMSADE UMR 7243 75775 Paris Cedex 16, France. fabio.furini@dauphine.fr 2Laboratoire d’Informatique de Paris Nord, Universit´e de Paris 13, Sorbonne Paris Cit´e, 99, Avenue J.-B. Clement 93430 Villetaneuse, France.
emiliano.traversi@lipn.univ-paris13.fr
Abstract
We perform a theoretical and computational study of the classical Linearisation Techniques (LT) and we propose a new LT for Binary Quadratic Problems (BQPs). We discuss the relations between the Linear Programming (LP) relaxations of the considered LT for generic BQPs. We prove that for a specific class of BQP all the LTs have the same LP relaxation value. We also compare the LT computational performance for four different BQPs from the literature. We consider the Unconstrained BQP and the Maximum Cut of edge-weighted graphs and, in order to measure the effects of constraints on the computational performance, we also consider the quadratic extension of two classical combinatorial optimization problems, i.e., the Knapsack and Stable Set problems.
keywords: Linearisation Techniques, Binary Quadratic Problems, Max Cut Problem, Quadratic Knapsack Problem, Quadratic Stable Set Problem.
1.
Introduction
A widely used technique for solving Binary Quadratic Problems (BQPs) is by formulating them as Mixed Integer Linear Programs (MILPs). The main advantage of these linear reformulations is the use of generic MILP solvers, which keep improving their computational performance (see for example Lodi [19]). Several Linearisation Techniques (LTs) have been proposed in the literature, the seminal works of this stream of research are Fortet [9] and Glover and Woolsey [12], both based on introducing additional binary variables. Other LTs have been proposed based only on introducing additional continuous variables; in this stream of research we cite the following works: Glover and Woolsey [13], Glover [11], and Sherali and Smith [24]. All these LTs are presented in detail in Section 2 and compared in Section 3, which is devoted to computational results. The LT of Chaovalitwongse et al. [6] is deeply related to the LT of Sherali and Smith and, for this reason, these two techniques will be discussed together. A further LT is presented by Adams and Sherali [2], this approach is generalized to design the reformulation–linearisation technique (RLT) in Sherali and Adams [23]. Recently, other interesting LTs have been introduced in Gueye and Michelon [14] and Hansen and Meyer [15]. In [14], the authors propose a framework for unconstrained BQP based on two steps: in the first step the objective function is decomposed with
a clique covering heuristic algorithm, while in the second step, valid inequalities are added to the model to get a better dual bound. In [15], a new family of LTs with a linear number of additional variables and constraints is presented. These LTs are based on a polyform representation of the objective function. The authors show that the best LT among the ones proposed is as tight as the LT of Glover and Woolsey [13] and it requires the solution of an auxiliary LP with a quadratic number of additional variables and constraints.
Paper Contribution. A first contribution of this paper is an overview of the strength of the Linear Programming (LP) relaxations of several LTs. Our analysis points out new interconnections between the classical LTs and an alternative LT, called Extended Linear Formulation (ELF). This new LT has been inspired by the ideas proposed in Jaumard et al. [17], where a reformulation specifically conceived for the Quadratic Stable Set Problem (QSSP) is proposed. For generic BQPs, the LT proposed by Glover and Woolsey in [13] and ELF are stronger than the LTs proposed by Glover in [11] and Sherali and Smith in [24]. The first two LTs require a quadratic number of additional variables and constraints, while the third and the forth require only a linear number.
A new theoretical result presented is the equivalence between the LP-relaxation value of all the LTs introduced for a specific class of BQPs. This class is rather generic and it contains the BQP formulation of the Max-Cut Problem.
An additional contribution of this paper is an extensive computational comparison of the LTs for four different classes of problem with and without linear constraints. In particular, we tested the Unconstrained BQP, the Max Cut Problem and the quadratic extensions of the Knapsack and the Stable Set problems.
2.
Linearisation techniques for Binary Quadratic Problems
A generic Binary Quadratic Problem (BQP), with n variables and p constraints, can be formulated using the following quadratic formulation:
min ( n X i=1 n X j=1 Qijxixj + n X i=1 Lixi : x ∈ K, x ∈ {0, 1}n ) ,
where Q ∈ Rn×n, L ∈ Rn, K = {x ∈ Rn : Ax ≥ b}, A ∈ Rp×n and b ∈ Rp. Q is a generic symmetric matrix, not restricted to being convex. Without loss of generality, we may assume Qii= 0, since we may incorporate those elements in the linear terms Li(for a given binary variable
x we have x2 = x). In the following, we define as the Linear Programming (LP) Relaxation of a formulation, the same formulation where the constraints x ∈ {0, 1}nare substituted by x ∈ [0, 1]n. In the next sections, we describe several alternatives for LTs for BQPs present in the literature and then we introduce the new Extended Linear Formulation (ELF).
2.1
Glover-Woolsey Linear Formulation
The standard method for linearising the quadratic terms is the one introduced by Glover and Woolsey and described in [13]. This linear formulation, called GW[13], reads as follows.
GW[13]: min n X i=1 n X j=i+1 2Qijyij + n X i=1 Lixi yij ≤ xi i, j = 1, . . . , n, i < j (1) yij ≤ xj i, j = 1, . . . , n, i < j (2) yij ≥ xi+ xj− 1 i, j = 1, . . . , n, i < j (3) yij ≥ 0 i, j = 1, . . . , n, i < j (4) x ∈ K x ∈ {0, 1}n.
A new variable yij takes the place of the product between the original variables xiand xj in the
objective function. We recall that GW[13]increases the size of the problem by adding n(n − 1)/2 variables and 4n(n − 1)/2 constraints. These constraints enforce yij = xixj for all couples of
binary variables xi and xj.
A constraint is redundant for a formulation if its removal does not change the set of optimal solutions of its LP-relaxation. Accordingly, the number of constraints of GW[13] can be reduced as follows:
Proposition 1. Inequalities (1) and (2), corresponding tonon-negativeentries ofQ, and inequal-ities(3) and (4), corresponding tonon-positiveentries ofQ, are redundant for GW[13].
Proof. See for example Forrester and Greenberg [8].
GW[13]can be strengthened by applying the so-called Reformulation Linearisation Technique (RLT) presented in Sherali and Adams [23]. The RLT is a procedure divided into two steps: the reformulation step creates additional nonlinear constraints by multiplying the constraints in K by product factors of the binary variables x and their complements (1 − x), and subsequently enforces the identity x2 = x. The linearisation step then substitutes a continuous variable for each distinct product of variables. Hence, GW[13]can be viewed as first level RLT applied only on the bound constraints 0 ≤ x ≤ 1. An alternative way to improve the strength of GW[13] is to include the so-called triangle and cycle inequalities (see for example [14] or [20]). In our analysis we do not consider these enhancements since we are interested in the basic version of GW[13].
2.2
Glover Linear Formulation
In this section we introduce the linearisation described in Glover [11]. This linear formulation, called G[11], reads as follows.
G[11]: min n X i=1 wi+ n X i=1 Lixi wi ≤ Q+i xi i = 1, . . . , n (5) wi ≥ Q−i xi i = 1, . . . , n (6) wi ≤ n X j=1 Qijxj− Q−i (1 − xi) i = 1, . . . , n (7) wi ≥ n X j=1 Qijxj− Q+i (1 − xi) i = 1, . . . , n (8) x ∈ K x ∈ {0, 1}n,
where Q+i and Q−i are suitable upper and lower bounds on the expressionPn
j=1Qijxj,
respec-tively. The main idea behind this linearisation is the introduction of the additional set of variables wi representing the quantityPnj=1Qijxj if xi = 1, or taking a value of 0 otherwise. Formulation
G[11]has the big advantage of using less variables and constraints than GW[13]. On the other hand, it requires the introduction of the so-called “big-M” constraints, leading to a weaker LP-relaxation. An important point concerns the computation of Q−i and Q+i . In Glover [11], the author sug-gests to impose: Q−i = n X j=1 min{0, Qij}, Q+i = n X j=1 max{0, Qij} . (9)
Subsequently, in several works (see for example Adams et al. [1] and Wang et al. [25]) this approach has been improved by taking into account the feasible region K and hence computing the values for Q−i and Q+i with more accuracy. This can be done as follows:
Q−i = min x∈K n X j=1 Qijxj, Q+i = maxx∈K n X j=1 Qijxj. (10)
It is easy to check that equations (9) and equations (10) coincide when K = ∅. From now on, when we refer to G[11]we imply the version with Q−i and Q+i defined in (9), since computing equations (10) may require the solution of an optimization problem which depends on the structure of K. Regardless of the method used for computing Q−i and Q+i , G[11] increases the size of the problem by adding only n variables and 4n constraints.
However, since we are minimizing and the coefficients of the w variables are positive, it is possible to reduce the number of constraints:
Proposition 2. Inequalities (5) and (7) are redundant for G[11]. Proof. See Adams et al. [1].
2.3
Sherali-Smith Linear Formulation
The third method linearises the quadratic terms using the techniques described in Sherali and Smith [24]. This linear formulation, called SS[24], reads as follows:
SS[24]: min n X i=1 si+ n X i=1 (Li+ Q−i )xi yi = n X j=1 Qijxj − si− Q−i i = 1, . . . , n (11) yi ≤ (Q+i − Q − i )(1 − xi) i = 1, . . . , n (12) si ≤ (Q+i − Q − i )xi i = 1, . . . , n (13) yi ≥ 0 i = 1, . . . , n (14) si ≥ 0 i = 1, . . . , n (15) x ∈ K x ∈ {0, 1}n ,
where Q+i and Q−i are defined as in (9) or in (10). Since the latter case requires the computation of an additional optimization problem, we use (9) as for G[11]. In Sherali and Smith [24], the authors introduce three formulations, called BP, ¯BP and BP-strong. However, in absence of quadratic constraints, they are all equivalent and they can be written as SS[24]. The idea behind SS[24]is similar to the one behind G[11], i.e., the introduction of additional variables representing the sumPn
j=1Qij − Q−i and using big-M constraints. SS[24]increases the size of the problem by
adding 2n variables, 4n inequalities and n equations. More precisely, variables y can be projected out using equations (11), thus resulting in only n additional variables and 4n inequalities (12)-(15). Formulation SS[24]is a strengthening of a precedent formulation proposed by Chaovalitwongse et al. [6] called CPP[6] which uses instead the following weaker (see Sherali and Smith [24]) definition of Q+i and Q−i : Q+i = max i n X j=1 |Qij|, Q−i = − maxi n X j=1 |Qij|.
Like for G[11], also for SS[24](and analogously for CPP[6]) some constraints can be eliminated: Proposition 3. Inequalities (13) and (14), are redundant for SS[24].
2.4
Extended Linear Formulation
We now introduce a new linearisation, called Extended Linear Formulation (ELF).
ELF : min n X i=1 n X j=1 Qij+ n X i=1 Lixi− n X i=1 n X j=i+1 2Qij(ziij + z j ij) ziji + zjij ≤ 1 i, j = 1, . . . , n, i < j (16) xi+ ziij ≤ 1 i, j = 1, . . . , n, i < j (17) xj+ z j ij ≤ 1 i, j = 1, . . . , n, i < j (18) xi+ ziij + z j ij ≥ 1 i, j = 1, . . . , n, i < j (19) xj+ ziji + z j ij ≥ 1 i, j = 1, . . . , n, i < j (20) x ∈ K x ∈ {0, 1}n .
This linearisation increases the size of the problem by adding n(n − 1) variables and 5n(n − 1) constraints. A pair of new variables ziij and zijj is used instead of the product of variables xi and
xj. These new variables modify the objective function and appear in the new set of constraints.
The total sum of the quadratic costs is paid in the objective function (Pn i=1
Pn
j=1Qij) and then
the correct value is reconstructed with the use of the z variables (Pn
i=1 Pn j=i+12Qij(z i ij + z j ij)).
The values of the z variables are set according to the values of the x variables thanks to constraints (16)-(20), i.e., if xi = xj = 1 then ziji = z
j
ij = 0, on the other hand, if one or both variables xiand
xj have a value of 0, then we have ziji + z j
ij = 1, which implies the correctness of the formulation.
Finally, the following Proposition allows the reduction of the number of constraints also for ELF:
Proposition 4. Inequalities (19) and (20), corresponding to non-negative entries of Q, and in-equalities(16), (17) and (18), corresponding to non-positive entries of Q are redundant for ELF.
Proof. Let (˜x, ˜zi, ˜zj) be an optimal solution of the LP-relaxation of ELF satisfying all inequalities
except one of the constraints (19) associated to an entry Qkl ≥ 0. Let xk+ zikl+ z j
kl ≥ 1 be the
violated inequality. And w.l.o.g, we assume that ˜zjkl ≥ 0, since the other constraints impose: 0 ≤ zi
kl+ z j
kl ≤ 1. We can increase the value of ˜ziklto ˜zkli + ∆ with ∆ = min{1 − ˜zkli − ˜z j
kl, 1 − ˜xk− ˜zkli }.
The constraint is violated hence ˜xk+ ˜zkli + ˜z j
kl< 1, this implies: (i) 0 < 1 − ˜z i kl− ˜z j kl, since 0 ≤ ˜xk and ˜xk< 1 − ˜zikl− ˜z j
kl; (ii) 0 < 1 − ˜xk− ˜zkli , since ˜xk+ ˜zikl < 1 and ˜z j
kl ≥ 0; and therefore, ∆ is
positive. By observing that ∆ corresponds to the minimum between the slacks of constraints (16) and (17) associated to k and l, the new solution is feasible by construction. The objective function value decreases by −∆Qkj This contradicts the fact of being optimal and violating one of the
constraints (19). The same holds for constraints (20).
Similar considerations can be done for inequalities (16), (17) and (18) when Qkl ≤ 0. Hence,
eliminating from ELF the inequalities (19) and (20), corresponding to a couple of indices k and l with non-negative Qklor the inequalities (16), (17) and (18), corresponding to a couple of indices
k and l with non-positive Qkl, does not change the set of optimal solutions of the LP-relaxation of
Formulations Variables Constraints Original Reduced GW[13] n(n − 1)/2 2n(n − 1) n(n − 1) G[11],CPP[6], SS[24] n 4n 2n ELF n(n − 1) 5n(n − 1) ≈ 5 2n(n − 1)
Table 1: Number of additional variables and constraints used in each LT.
Constraints (16), (17) and (18) provide upper bounds on the variables z, while constraints (19) and (20) provide lower bounds. Hence, according to the sign of the coefficients of the matrix Q and the sense of the objective function, some of them are redundant.
2.5
Formulation Summary
In Table 1, we report the number of additional variables and constraints needed by each LT. The columns concerning the constraints are subdivided in two parts: the table column – Original – report the number of inequalities needed by each formulation while the table column – Reduced – report the number of non redundant constraints needed by each formulation (see Propositions 1, 2, 3 and 4). In the following, this reduction of the formulation dimensions will be referred to as Constraint-Redundancy elimination. As far as GW[13], G[11], CPP[6] and SS[24] are concerned, regardless of the sign of the entries of Q, the Constraint - Redundancy elimi-nation allows to reduce by one half the number of additional constraints. For ELF the situation is slightly different because the number of non redundant constraints depends on the sign of Q and it is hence included in the interval [2n(n − 1), 3n(n − 1)], for this reason we reported the average value of 52n(n − 1). As explained in Adams et al. [1], the number of constraints of G[11]can be further reduced to n via a variable transformation.
2.5.1 Formulation strength
In this section, we present a comparison of the strength of the LT discussed in the previous sections.
The strong connection between G[11]and SS[24]is given by the following proposition:
Proposition 5. The polyhedra defined by the LP-relaxations of G[11]and SS[24]are isomorphic under a linear transformation that keeps the values of thex variables unchanged and the relation between thes and the w variables is the following:
si = wi− Q−i xi i = 1, . . . , n.
Proof. As first step, it is possible to apply to SS[24]the variable substitution si = wi − Q−i xi and
subsequently equations (11) can be used to substitute variables y into the remaining constraints, thus obtaining exactly G[11]. By applying to G[11] the variable substitution wi = si+ Q−i xi, we
A direct consequence of Proposition 5 is that the optimal LP-relaxation values of G[11] and SS[24]are identical.
The following proposition shows that ELF is an extended formulation of GW[13]:
Proposition 6. The polyhedra defined by the LP-relaxations of GW[13] andELF are isomorphic under a linear transformation that keeps the values of thex variables unchanged and the relations between thez and the y variables are the following:
ziij = 1 − xi, zijj = xi− yij i, j = 1, . . . , n, i < j.
Proof. Let (¯x, ¯y) be a feasible solution to the LP-relaxation of GW[13]. We consider the following solution (˜x, ˜zi, ˜zj) of the LP-relaxation of ELF: ˜x
i = ¯xi, ˜ziji = 1 − ¯xi, ˜zijj = ¯xi− ¯yij. It is easy to
check that it is feasible and it has the same value. Now let (˜x, ˜zi, ˜zj) be a feasible solution to the LP-relaxation of ELF. We consider the following solution (¯x, ¯y) of the LP-relaxation of GW[13]: ¯
xi = ˜xi, ¯yij = 1 − ˜ziji − ˜z j
ij. Again, it is easy to check that it is feasible and it has the same
value. This implies that for every feasible solution of the LP-relaxation of GW[13] there exists one feasible solution of the LP-relaxation of ELF of the same value and vice versa, proving the statement.
Also in this case, a direct consequence of Proposition 6 is that the LP-relaxation values of GW[13] and ELF are identical. The new linearisation ELF can be seen as an extended formulation (see e.g., [7]) of GW in the sense that the projection on the (x,y)-space of the polytope defined by (16)-(20), and constraints yij = 1−ziji −z
j
ij (i, j = 1, . . . , n, i < j), is equal to the polytope defined
by (1)-(4) and by x ∈ [0, 1]n. Moreover, Proposition 6 also implies that every valid inequality for
GW[13]is also valid for ELF.
We recall that in the following analysis we use formulas (9), which imply:
n X j=1 Qij = Q+i + Q − i .
To the best of our knowledge, no formal proof concerning the relation between the LP relaxations of GW[13] and G[11] has been given in the literature. For sake of completeness, we present the following Proposition, that will be useful in the rest of the section:
Proposition 7. The LP-relaxation of GW[13]is stronger than the LP relaxation of G[11].
Proof. Let (˜x, ˜y) be a feasible solution of the LP-relaxation of GW[13], we consider the solution (¯x, ¯w) obtained from (˜x, ˜y) as follows:
¯ xi = ˜xi w¯i = n X j=1 Qijy˜ij i = 1, . . . , n;
both solutions have by construction the same objective function value. We now show that the solution (¯x, ¯w) is feasible for the LP-relaxation of G[11]. We notice that, for each i = 1, . . . , n, we
have: ¯ wi = n X j=1 Qijy˜ij = X j=1...,n : Qij<0 Qijy˜ij + X j=1...,n : Qij≥0 Qijy˜ij ≥ X j=1...,n : Qij<0 Qijy˜ij ≥ X j=1...,n : Qij<0 Qijx˜i = Q−i x˜i
where the last inequality derives from ˜yij ≤ ˜xi. Analogously, for each i = 1, . . . , n, we have:
¯ wi = n X j=1 Qijy˜ij = X j=1...,n : Qij<0 Qijy˜ij + X j=1...,n : Qij≥0 Qijy˜ij ≥ ≥ X j=1...,n : Qij<0 Qijx˜j + X j=1...,n : Qij≥0 Qijx˜i+ X j=1...,n : Qij≥0 Qijx˜j − X j=1...,n : Qij≥0 Qij = = n X j=1 Qijx˜j− Q+i (1 − ˜xi)
where the first inequalities derive from ˜yij ≤ ˜xi and ˜yij ≥ ˜xi + ˜xj − 1. The two sequences
of inequalities show that the solution (¯x, ¯y) respects inequalities (6) and (8). Finally, we provide an instance where the optimal value of the LP-relaxation of GW[13] is greater than the optimal value of the LP-relaxation of G[11]. Consider the following instance with K = ∅, n = 3 and
L = 1 −1 8 ; Q = 0 −1 1 −1 0 −5 1 −5 0
. The optimal solution of the LP-relaxation of GW[13]
is ˜x1 = 0, ˜x2 = 1, ˜x3 = 1, ˜y12 = 0, ˜y13 = 0, ˜y23 = 1 and the optimal solution value is −3. On
the other hand, the optimal solution of the LP-relaxation of G[11]is ¯x1 = 0.54, ¯x2 = 1, ¯x3 = 0.90,
¯
w1 = −0.54, ¯w2 = −5.09, ¯w3 = −4.54 and the optimal solution value is −3.36.
We now introduce two lemmas that are used to show a new property of the LP-relaxation of GW[13]and G[11]. Both lemmas use the following Proposition:
Proposition 8. In case of K = ∅, given a feasible vector ¯x for GW[13](resp. G[11]), the optimal values of they (resp. ¯¯ w) are:
¯ yij = max{0, ¯xi+ ¯xi− 1} ifQij ≥ 0 min{¯xi, ¯xj} ifQij < 0 i, j = 1, . . . , n, ¯ wi = max ( Q−i x¯i, n X j=1 Qijx¯j− Q+i (1 − ¯xi) ) i = 1, . . . , n.
Proof. The statement is a direct consequence of the sign of the y and w variables in the respective objective functions and of the fact that there are no additional constraints, except for the ones linking the x variables to the y and w variables respectively.
Lemma 1. If K = ∅, let (˜x, ˜y) and (¯x, ¯w) be two optimal solutions of the LP-relaxations of GW[13] and G[11]respectively. Ifx˜i = ¯xi =
1
2 for alli = 1, . . . , n, both solutions have the same objective function value equal to 1
2 Pn
i=1(Q − i + Li).
Proof. It is sufficient to compute the objective function value of both solutions. If Qij ≥ 0, we
have ˜yij = max{0, ˜xi + ˜xi − 1} = max{0, 0} = 0. If Qij < 0, we have ˜yij = min{˜xi, ˜xj} =
min 1 2, 1 2 = 1
2. The objective function value of GW[13]is hence the following: n X i=1 n X j=1 Qijy˜ij + n X i=1 Lix˜i = 1 2 X i,j=1...,n : Qij<0 Qij + 1 2 n X i=1 Li = 1 2 n X i=1 (Q−i + Li);
Moreover, we have ¯wi = max{Q−i xi,
Pn
j=1Qijx¯j−Q +
i (1− ¯xi)}, leading to the following objective
function value of G[11]: n X i=1 ¯ wi+ n X i=1 Lix¯i = n X i=1 max ( Q−i x¯i, n X j=1 Qijx¯j − Q+i (1 − ¯xi) ) + n X i=1 Lix¯i = n X i=1 max ( 1 2Q − i , 1 2 n X j=1 (Qij − Q+i ) ) + 1 2 n X i=1 Li = n X i=1 max 1 2Q − i , 1 2Q − i + 1 2 n X i=1 Li = 1 2 n X i=1 (Q−i + Li). Lemma 2. If K = ∅ and Li = − Pn
j=1Qij (i = 1, 2, . . . , n), an optimal solution of the
LP-relaxation of GW[13]is the following:
¯ xi = 1 2, i = 1, . . . , n, ¯ yij = ( 0 ifQij ≥ 0; 1 2 ifQij < 0 i, j = 1, . . . , n.
Proof. The objective function is separable, hence, we measure the impact of changing one single variable at a time. We consider the solutions ˆx = ¯x ± eˆiwith ˆi = 1, . . . , n and ≥ 0, where eˆi is
an n-dimensional vector with value 1 in the ˆi-th component and 0 otherwise.
We consider first the case ˆx = ¯x + eˆi. Proposition 8 implies that the ˆy variable corresponding to
ˆ
x = ¯x + eˆi is equal to ¯y, except for i = ˆi, where we have:
ˆ yˆij = ( if Qˆij ≥ 0 1 2 if Qˆij < 0 j = 1, . . . , n.
ˆi: n X i=1 n X j=1 Qijyˆij + n X i=1 Lixˆi= n X i=1 n X j=1 Qijyˆij − n X i=1 ( n X j=1 Qij)ˆxi = = n X i=1 n X j=1 Qijy¯ij + X j=1...,n : Qˆij≥0 Qˆij − n X i=1 ( n X j=1 Qij)¯xi− X j=1...,n : Qˆij≥0 Qˆij − X j=1...,n : Qˆij<0 Qˆij = = n X i=1 n X j=1 Qijy¯ij − n X i=1 ( n X j=1 Qij)¯xi− X j=1...,n : Qˆij<0 Qˆij ≥ n X i=1 n X j=1 Qijy¯ij + n X i=1 Lix¯i.
Secondly, if we consider the solution ˆx = x − eˆi, it also differs from ¯x when i = ˆi, where we
have: ¯ yˆij = ( 0 if Qˆij ≥ 0 1 2 − if Qˆij < 0 i, j = 1, . . . , n. This leads to n X i=1 n X j=1 Qijyˆij + n X i=1 Lixˆi= n X i=1 n X j=1 Qijyˆij − n X i=1 ( n X j=1 Qij)ˆxi = = n X i=1 n X j=1 Qijy¯ij − X j=1...,n : Qˆij<0 Qˆij − n X i=1 ( n X j=1 Qij)¯xi+ X j=1...,n : Qˆij≥0 Qˆij + X j=1...,n : Qˆij<0 Qˆij = = n X i=1 n X j=1 Qijy¯ij − n X i=1 ( n X j=1 Qij)¯xi+ X j=1...,n : Qˆij≥0 Qˆij ≥ n X i=1 n X j=1 Qijy¯ij + n X i=1 Lix¯i.
In both cases, the objective function value of (ˆx, ˆy) is greater than the one of (¯x, ¯y), proving that (¯x, ¯y) is optimal.
The following theorem introduces an interesting class of BQP instances, where all the LT considered in this paper are equivalent in terms of LP-relaxation:
Theorem 1. If K = ∅ and Li= −
Pn
j=1Qij(i = 1, 2, . . . , n), the optimal solution values of the
LP-relaxation of GW[13]and G[11]coincide and they are both equal to 1 2
Pn
i=1(Q −
i + Li). Moreover,
let (¯x, ¯w) and (˜x, ˜y) be the optimal solutions of their LP-relaxations, we have ¯xi = ˜xi =
1 2 (i = 1, . . . , n) and ˜y and ¯w are defined accordingly to Proposition 8.
Proof. If Li= −
Pn
j=1Qij, we can apply Lemma 2 to identify the optimal solution of the
LP-relaxation of GW[13] and its value. We can hence use Lemma 1 to show that there exists an optimal solution to G[11] with ¯x = ˜x and with the same objective function value. Finally, we notice that, G[11]can not be stronger than GW[13](Proposition 7), for this reason, we have that the solution with ¯x = ˜x = 1
LP of GW[13] ⇒ (P rop. 7) LP of G[11] m (P rop. 6) m (P rop. 5)
LP of ELF LP of SS[24] ⇒ (see [24]) LP of CPP[6]
Table 2: Summary of the relations between the strength of the LP-relaxations.
An important class of instances, satisfying Theorem 1, corresponds to BQP reformulation of the Max Cut problem (see Section 3.1).
Finally, in Table 2, we summarize the relations between the different LP-relaxations of the formulations studied. The relation “A ⇔ B” stands for “A and B have the same LP-relaxation value” and “A ⇒ B” stands for “A has a stronger LP-relaxation value than B”. For each relation, we report the reference where it is proved.
3.
Computational experiments
In this section, we assess the computational performances of the LTs discussed in this paper, i.e., GW[13], G[11], SS[24]and the new ELF. Preliminary experiments showed that the LP-relaxation of CPP[6] is extremely weak on all the test-beds, this is due to the fact of having large big-M values compared to the ones of SS[24]. Since this formulation is computationally dominated by SS[24], it is dropped from the comparison tests.
The first part of our tests is based on the Unconstrained BQP and on the Maximum Cut Prob-lem. The second part of our tests is based on quadratic instances with linear constraints, more precisely we consider the Quadratic Knapsack Problem and the Quadratic Stable Set Problem. In our computational comparison we decided not to use any problem dependent valid inequalities, since our goal is to compare the formulations in their basic versions.
All the experiments have been performed on a computer with a 3.40 Ghz 8-core Intel Core i7-3770 processor and 16Gb RAM, running a 64 bits Linux operating system. We use CPLEX 12.7.0 [16] as MILP solver ran single-threaded with default parameter settings. The tests have been performed with the Constraint-Redundancy elimination activated. It is worth mentioning that, thanks to preliminary experiments, we observed a substantial worsening (up to several orders of magnitude) of the overall computing time of CPLEX when this intuitive adjustment of the LTs is not applied.
MILP solvers implement different branch-and-cut algorithms and they all heuristically separate generic cuts in order to improve the quality of the dual bounds. In addition, the solvers perform several variable-and-constraint reduction techniques which are based on the specific formulation to solve. These aspects must be taken into account when analyzing the computational performance of different LTs. For example a more aggressive strategy of a solver applied to a specific LT (rather than to another one) may influence the computational behavior. Unfortunately these distortions generated by the specific implementations of the solvers are impossible to eliminate. In other words, the computational analysis of the next section is dependent on the choice of CPLEX 12.7.0 and the conclusions may be slightly different in case an other solver is used (or simply other settings of parameters). Nevertheless our experiments give an overall assessment of the relative
computational performance of the different LTs for four different classes of problems and using one of the state-of-the-art MILP solvers.
3.1
Unconstrained Problems
Problem description. Our first aim is to investigate the strength of the different formulations without the influence of additional constraints (K = ∅). For this reason, we adopt the Biq Mac library (see Wiegele [26]) as a case study. The Biq Mac library is a collection of instances widely used in the literature, see for example Rendl et al. [22] or, more recently, Krislock et al. [18]. It is composed of two families of problems: the first one is the Unconstrained BQP (UBQP):
min ( n X i=1 n X j=1 Qijxixj+ n X i=1 Lixi : x ∈ {0, 1}n ) ,
where Q is a symmetric matrix of order n. The second family is the Max Cut (MC) problem. The MC consists of finding a maximum weighted bipartition of a graph G of n vertices (see for example Rendl et al. [22]) and it can be formulated as follows:
max ( n X i=1 n X j=1 ˜ Qijxixj : x ∈ {−1, 1}n ) ,
where ˜Q is the Laplacian matrix of the graph G. It is well known that the UBQP and the MC are equivalent (see Caprara [4] or Krislock et al. [18]).
The Biq Mac library used for the experiments is divided into five classes: beasley, gka, and beare UBQP instances, while rudy, ising are MC instances. The classes altogether form a test bed of 343 instances. Some are randomly generated instances and others come from a statistical physics application. In each class, the instances differ in terms of size n, density of the matrix Q and L (for further details on the instance features, we refer the reader to Wiegele [26]).
In our experiments we used the UBQP formulation of the MC problem obtained with the fol-lowing transformation. Let wij (i, j = 1, . . . , n, wij = wji) be the set of weights associated to a
given graph G, the UBQP for the MC on G has the following objective function (see [4]):
1 2 n X i=1 n X j=1 wij[xi(1 − xj) + xj(1 − xi)] = − n X i=1 n X j=1 wijxixj + 1 2 n X i=1 n X j=1 wij(xi+ xj) = − n X i=1 n X j=1 wijxixj+ 1 2 n X i=1 n X j=1 wijxi + 1 2 n X i=1 n X j=1 wijxj = − n X i=1 n X j=1 wijxixj + n X i=1 n X j=1 wijxi .
The last term of the chain of equations shows that all UBQP instances arising from MC instances have Li = −
Pn
j=1Qij and hence belong to the class of instances of Theorem 1.
Tables description. In Table 3 and Table 4 we report the results concerning the whole Biq Mac Library. Both tables are divided into 5 horizontal blocks, one for each class of instances (be,
LP times LP iterations LP gaps
n # GW[13] ELF G[11] SS[24] GW[13] ELF G[11] SS[24] GW[13] ELF G[11] SS[24]
be 100 10 0.07 0.31 0.01 0.01 4292.1 8831.0 210.5 100.0 73.6 73.6 73.6 73.6 120 20 0.05 0.22 0.01 0.01 3451.2 7033.0 259.7 122.4 63.6 63.6 63.7 63.7 150 20 0.27 0.48 0.02 0.01 4899.6 10985.4 328.2 151.5 69.1 69.1 69.1 69.1 200 20 0.82 1.50 0.04 0.02 8669.5 19494.1 437.4 200.6 79.1 79.1 79.1 79.1 250 10 0.07 0.11 0.04 0.03 2798.4 5856.1 569.6 291.5 42.6 42.6 43.7 43.7 beasl ey 100 1050 10 0.000.01 0.000.01 0.000.00 0.000.00 120.9676.1 1034.4234.2 218.082.0 136.563.4 12.9 12.90.1 0.1 22.910.2 10.222.9 250 10 0.07 0.12 0.05 0.03 2822.2 5943.2 598.8 278.8 44.3 44.3 44.9 44.9 500 10 0.64 1.11 0.32 0.19 10543.1 22787.4 1184.8 504.1 87.4 87.4 87.4 87.4 g k a 20 1 0.00 0.00 0.00 0.00 28.0 408.0 20.0 21.0 66.0 66.0 67.3 67.3 30 3 0.00 0.00 0.00 0.00 171.7 561.3 54.3 34.7 40.3 40.3 43.1 43.1 40 2 0.00 0.02 0.00 0.00 337.5 1382.5 62.5 41.0 60.4 60.4 61.2 61.2 50 4 0.00 0.01 0.00 0.00 312.3 1143.0 84.0 58.5 34.3 34.3 39.4 39.4 60 3 0.00 0.01 0.00 0.00 328.7 1738.3 91.3 71.3 38.3 38.3 41.7 41.7 70 3 0.01 0.02 0.00 0.00 390.3 2252.0 122.7 81.7 40.1 40.1 45.8 45.8 80 3 0.00 0.02 0.00 0.00 465.7 2779.0 133.0 98.0 35.1 35.1 43.1 43.1 90 2 0.01 0.04 0.00 0.00 341.0 4443.0 131.5 108.5 45.7 45.7 52.7 52.7 100 13 0.02 0.10 0.00 0.00 1925.1 4654.2 203.5 109.3 50.7 50.7 53.3 53.3 125 1 0.03 0.15 0.01 0.01 182.0 15366.0 118.0 121.0 92.2 92.2 92.7 92.7 200 5 0.27 0.60 0.03 0.02 4898.2 10759.0 431.6 209.6 58.9 58.9 59.7 59.7 500 5 32.52 64.78 0.55 0.35 50312.4 114504.4 1153.4 500.4 92.8 92.8 92.8 92.8 ising 100 9 0.03 0.07 0.01 0.00 3437.1 6506.9 203.7 101.1 20.4 20.4 20.4 20.4 125 3 0.01 0.01 0.00 0.00 510.0 842.7 261.7 127.3 27.2 27.2 27.2 27.2 150 6 0.13 0.29 0.02 0.01 10253.0 19422.7 306.8 150.0 19.2 19.2 19.2 19.2 200 6 0.24 0.57 0.03 0.02 15686.3 29183.7 409.2 200.0 18.8 18.8 18.8 18.8 216 3 0.01 0.02 0.01 0.01 777.0 1438.3 456.3 217.7 29.1 29.1 29.1 29.1 225 3 0.00 0.02 0.01 0.01 633.0 1082.7 462.0 234.7 16.9 16.9 16.9 16.9 250 6 0.32 0.95 0.05 0.04 20947.0 39318.8 512.5 250.0 19.9 19.9 19.9 19.9 300 6 0.56 1.15 0.08 0.07 26308.3 49666.8 616.3 300.0 19.5 19.5 19.5 19.5 343 3 0.02 0.04 0.03 0.02 1233.3 2283.3 722.7 346.3 29.7 29.7 29.7 29.7 400 3 0.02 0.04 0.02 0.02 1117.0 1936.7 817.0 415.7 17.9 17.9 17.9 17.9 r udy 60 10 0.00 0.02 0.00 0.00 618.9 2334.2 63.2 63.2 40.1 40.1 40.1 40.1 80 30 0.01 0.06 0.00 0.00 1372.1 3478.5 136.2 81.0 59.0 59.0 59.0 59.0 100 90 0.02 0.10 0.00 0.00 1916.5 5402.2 161.7 102.6 56.6 56.6 56.6 56.6
MILP times MILP branching nodes exit gaps
n # GW[13] ELF G[11] SS[24] GW[13] ELF G[11] SS[24] GW[13] ELF G[11] SS[24]
be 100 10 tl 0 tl 0 tl 0 tl 0 903 159877 489570 294827 33.8 64.7 50.0 51.1 120 20 1730.6 1 tl 0 tl 0 tl 0 4400 368818 286643 218557 20.5 50.4 40.0 40.8 150 20 tl 0 tl 0 tl 0 tl 0 1667 234478 229528 162299 33.2 60.2 47.3 50.5 200 20 tl 0 tl 0 tl 0 tl 0 266 117224 129028 207618 57.2 69.0 54.7 62.1 250 10 tl 0 tl 0 tl 0 tl 0 4032 433728 41637 42396 8.5 32.7 25.5 25.3 beasl ey 100 1050 10 0.0 100.3 10 0.0 100.2 10 11.2 100.0 10 0.0 107.2 10 05 1420 163774 14401 0.00.0 0.00.0 0.00.0 0.00.0 250 10 tl 0 tl 0 tl 0 tl 0 3642 484278 41674 47243 10.1 35.4 27.4 27.5 500 10 tl 0 tl 0 tl 0 tl 0 0 75038 6188 86809 81.7 58.9 54.4 59.2 g k a 20 1 0.0 1 0.0 1 0.0 1 0.0 1 0 21 44 43 0.0 0.0 0.0 0.0 30 3 0.0 3 0.0 3 0.1 3 0.1 3 0 33 247 109 0.0 0.0 0.0 0.0 40 2 0.7 2 0.4 2 1.8 2 1.8 2 0 363 2222 1324 0.0 0.0 0.0 0.0 50 4 0.6 4 1.0 4 10.2 4 8.7 4 2 771 4607 3771 0.0 0.0 0.0 0.0 60 3 0.5 3 0.5 3 5.2 3 4.6 3 9 375 1153 1460 0.0 0.0 0.0 0.0 70 3 0.8 3 0.6 3 7.1 3 7.7 3 12 266 1927 1731 0.0 0.0 0.0 0.0 80 3 0.5 3 0.5 3 4.9 3 3.9 3 12 160 1466 887 0.0 0.0 0.0 0.0 90 2 0.5 2 1.0 2 1.6 2 1.1 2 30 141 784 532 0.0 0.0 0.0 0.0 100 13 987.5 6 1241.4 4 1242.1 4 1241.5 4 2848 332199 264265 208236 9.9 30.1 24.6 25.4 125 1 3.8 1 6.9 1 5.0 1 7.6 1 176 529 3170 2827 0.0 0.0 0.0 0.0 200 5 1465.3 1 tl 0 tl 0 tl 0 1079 211712 118335 114323 24.9 50.1 39.9 44.9 500 5 tl 0 tl 0 tl 0 tl 0 0 19242 5329 130956 90.1 86.3 75.3 78.0 ising 100 9 74.2 9 65.6 9 1644.5 1 1699.4 1 5679 10918 1943960 3449968 0.0 0.0 7.6 7.5 125 3 1.6 3 17.9 3 tl 0 tl 0 38 27326 3487383 4886222 0.0 0.0 15.2 16.4 150 6 902.3 4 1045.3 3 tl 0 tl 0 16394 55880 370536 1059850 1.0 1.3 13.3 12.5 200 6 1697.4 1 tl 0 tl 0 tl 0 30829 98811 265229 647454 4.5 4.3 14.8 14.0 216 3 13.6 3 tl 0 tl 0 tl 0 595 1896661 3025926 2521665 0.0 7.3 23.9 23.0 225 3 0.8 3 4.9 3 tl 0 tl 0 57 6695 3973997 3392715 0.0 0.0 11.9 11.8 250 6 tl 0 tl 0 tl 0 tl 0 28752 66611 202984 413173 7.8 7.6 17.1 16.3 300 6 tl 0 tl 0 tl 0 tl 0 27969 58156 161895 282489 8.3 8.9 17.2 16.3 343 3 140.4 3 tl 0 tl 0 tl 0 2496 1124983 1645263 1550100 0.0 15.2 27.0 25.6 400 3 2.4 3 900.7 3 tl 0 tl 0 162 594238 1580827 1505487 0.0 0.0 14.1 13.3 r udy 60 10 tl 0 tl 0 tl 0 tl 0 44880 1897854 3508202 1294517 3.8 19.8 17.0 17.4 80 30 1197.4 10 1209.3 10 tl 0 tl 0 4691 503775 1005699 717697 21.2 35.1 33.4 33.0 100 90 1223.8 30 1561.2 20 tl 0 tl 0 3897 676430 784450 533997 22.7 37.5 35.3 33.7
Table 4: Computational comparison of the different formulations for solving Biq Mac instances to proven optimality
beasly, gka, ising and rudy). In each line we group together the results relative to instances of the same size (same values of n). The first two columns of both tables report the size of the subclasses of instances and their cardinality (i.e., the number of instances of a specific subclass).
In Table 3 we report the results concerning the computational behavior for solving the LP-relaxation of the LTs considered. The table provides the following information:
• LP times. The average time required by CPLEX to solve the LP-relaxation to proven opti-mality.
• LP iterations. The average number of simplex iterations need by CPLEX to solve the LP-relaxation.
• LP gaps. The average gap (in percentage) between the optimal value of the LP-relaxation and the optimal solution value (or best know incumbent solution).
In Table 4 we report the results concerning the computational behavior for solving the test problems with the different LTs to proven optimality. The table provides the following information:
• MILP times. The average computational time required by CPLEX to solve the problem to proved optimality and the number of instances solved within the time-limit of 1800 seconds (we report tl and 0 in case all the instances of a specific subclass and formulation reach the time limit). In each line, the most efficient LT is reported in bold text, i.e., the one with shorter computing time or larger number of instances solved respectively.
• MILP branching nodes. The average number of nodes explored by CPLEX in the Branch-and-Bound tree.
• MILP exit gaps. The average percentage exit gap between the upper and the lower bounds computed by CPLEX in case the time limit is reached (0.0 is reported otherwise). The LT with the lowest open gap is reported in bold text.
Tables discussion. Table 3 shows, as expected, that the four LTs considered present the same LP-relaxation values for the MC instances. On the other hand, the situation for the Biq instances is different: GW[13] and ELF are characterized by stronger LP-relaxation values (the difference is less clear for the be and beasley class of instances). As far as LP times and iterations are concerned, GW[13]and ELF take a higher number of iterations and longer computing times than G[11] and SS[24]. This figure can be explained by the higher number of additional variables and constraints. Since for the MC instances all the LP-relaxation bounds are equivalent, SS[24]is the more effective option to compute it.
Concerning Table 4, the formulations with the best computational behavior for the Biq Mac in-stances are GW[13]and ELF. These formulations clearly outperform G[11]and SS[24]concerning both the average computing time and the total number of instances solved. For the Biq instances, this is due to the fact of having stronger LP-relaxation values which allows a better computational convergence. For the Mac instances, where the LP-relaxation values coincides, the different behav-ior can be explained by the better performance of GW[13]and ELF during the branching scheme and by the efficacy of the generic cuts generated by CPLEX. In case the time limit is reached, ELF explores a higher number of branching nodes than GW[13] but the average exit gaps are higher. This figure is due to the different policy of CPLEX in separating cuts at the root node, i.e., for
GW[13], CPLEX spends a large amount of time in improving the root node bound, while for ELF CPLEXstarts immediately to branch.
3.2
Constrained Problems
In the remaining part of this computational section we introduce and discuss the other two test problems considered, i.e., the Quadratic Stable Set Problem and the Quadratic Knapsack Problem. These two problems are used to measure the behaviour of the different LTs discussed in this paper in presence of constraints.
3.2.1 Quadratic Stable Set Problem
Problem description. We recall that a stable set is a subset of fully disconnected vertices. Fol-lowing the notation of Section 2, we present the problem in minimization form. The formal def-inition of the Quadratic Stable Set Problem (QSSP) is the following: given an undirected graph G = (V, E), with V = {v1, . . . , vn} the set of vertices and E the set of edges, a vector of linear
cost L ∈ Rn on the vertices and a symmetric matrix of quadratic costs Q ∈ Rn×n on couples of vertices, the QSSP searches for a stable set of G with minimum cost. In other words, if vertices i and j are in the solution, not only the linear costs are collected but also an additional quadratic cost equal to Qij is considered.
The mathematical formulation of the QSSP reads as follows:
min ( n X i=1 n X j=i+1 2Qijxixj+ n X i=1 Lixi : xi+ xj ≤ 1, ∀(vi, vj) ∈ E, x ∈ {0, 1}n )
This quadratic counterpart of the Linear Stable Set Problem has not received much attention in the literature (for further details on the QSSP we refer the interested reader to Furini and Traversi [10]). As test-bed we used 16 instances introduced in [10], based on random graphs a number of vertices n = 150, density µ ∈ {50%, 75%}, and a percentage of negative quadratic costs ν ∈ {25%, 50%, 75%}. We used 2 instances for each class using different random seeds. In the following paragraph we discuss the special case of QSSP with ν = 0%
A relevant special case. As observed in Jaumard et al. [17], the QSSP with only non-negative costs, i.e., Qij ≥ 0, vi, vj ∈ V , can be viewed as a weighted Stable Set Problem (SSP) on an
extended graph ˜G = ( ˜V , ˜E), where ˜V = V ∪ V1∪ V2, ˜E = E ∪ E1∪ E2∪ E12and
V1 = {viji : vi, vj ∈ V, Qij > 0}, V2 = {v j ij : vi, vj ∈ V, Qij > 0}, E1 = {(vi, viij), vi ∈ V, viji ∈ V1 : Qij > 0}, E2 = {(vijj, vj), vj ∈ V, vijj ∈ V2 : Qij > 0}, E12= {(viji , v j ij), v i ij ∈ V1, vijj ∈ V2 : Qij > 0}.
4 6 8 2 2 2 v1, −15 v2, −15 v3, −2 v4, −10 v5, −8 v6, −8 v1, −15 v2, −15 v3, −2 v4, −10 v5, −8 v6, −8 v2 26, −4 v2 23, −6 v3 23, −6 v3 35, −8 v5 35, −8 v5 56, −2 v6 56, −2 v6 26, −4 v5 25, −2 v6 36, −2 v3 36, −2 v2 25, −2
Figure 1: Example: a graph G (left) and the corresponding extended graph ˜G (right).
The cost of nodes viji ∈ V1and vijj ∈ V2are equal to −Qij, replacing in this way the initial quadratic
costs Q. The optimal solution value of QSSP on G is then equal to the optimal solution value of the weighted SSP on ˜G plus the constantPn
i=1
Pn
j=1Qij.
An example of the extended graph is given in Figure 1. On the left part of the figure we report a graph G = (V, E) with 6 vertices and 9 edges and the following costs:
Q = 0 0 0 0 0 0 0 0 3 0 1 2 0 3 0 0 4 1 0 0 0 0 0 0 0 1 4 0 0 1 0 2 1 0 1 0 ; L = −15 −15 −2 −10 −8 −8 .
The dashed lines represent a positive entry Qij. On the right part of the figure we report the
corresponding extended graph ˜G = ( ˜V , ˜E) with 18 vertices and 25 edges. In both graph, the optimal solutions are represented by the vertices in red. The cost of the stable set {v2, v5, v6} in
G is equal to −23, while the cost of the stable set {v2, v5, v6, v233 ; v353 , v363 } in ˜G is equal to −47.
Finally, the constant part is equal to 24.
In other words, for the QSSP with Qij ≥ 0, ELF introduces only the sets of constraints (16),
(17) and (18) and, hence, it is equivalent to the standard ILP formulation with edge constraints for the Weighted Stable Set Problem on the extended graph ˜G.
Table description. In Table 5 we report the results concerning the computational behaviour of the four LTs studied on the QSSP benchmark. In each line, we group together the average results relative to instances with same values of µ and ν. Each entry of the table provides the aggregated results on 2 instances (# = 2 on the table). We provide, for each LT, the following information:
• MILP times. The average computational time required by CPLEX to solve the problem to proved optimality.
MILP times MILP branching nodes n µ ν # GW[13] ELF G[11] SS[24] GW[13] ELF G[11] SS[24] 150 50 25 2 139.68 340.94 216.82 334.01 1613.50 3143.00 198666.50 126996.00 50 2 1216.65 540.30 291.52 366.40 3915.50 10166.50 337011.00 171489.00 75 2 1490.78 626.17 314.91 362.90 14382.00 15674.50 394338.50 169204.00 75 25 2 37.59 52.26 19.29 44.41 149.50 139.00 6277.00 7910.00 50 2 85.14 84.61 20.70 44.03 260.50 264.00 9651.00 7731.50 75 2 98.99 119.84 23.99 43.58 832.50 835.50 13440.50 10856.00 LP times LP gaps n µ ν # GW[13] ELF G[11] SS[24] GW[13] ELF G[11] SS[24] 150 50 25 2 0.05 0.49 0.07 0.08 98.14 98.14 99.08 99.08 50 2 0.06 0.86 0.02 0.06 98.06 98.06 99.01 99.01 75 2 0.07 1.28 0.01 0.02 98.00 98.00 98.98 98.98 75 25 2 0.01 0.41 0.01 0.12 97.47 97.47 99.41 99.41 50 2 0.01 0.59 0.01 0.08 97.68 97.68 99.42 99.42 75 2 0.01 0.05 0.01 0.03 98.41 98.41 99.59 99.59
Table 5: Computational comparison of the different formulations for solving QSSP instances
• MILP branching nodes. The average number of nodes in the Branch-and-Bound tree ex-plored by CPLEX.
• LP times. The average time required by CPLEX to solve the LP-relaxation to proven opti-mality.
• LP gaps. The average percentage gap between the optimal value of the LP-relaxation and the optimal solution.
Table discussion. The formulations with the best computational behavior for the QSSP instances are G[11]and SS[24]on average. ELF presents on average a better computing time of GW[13]The number of MILP branching nodes of ELF is slightly greater than GW[13]. G[11] and SS[24] are characterized by a higher number of nodes. This fact is not surprising since those formulations have a weaker LP-relaxation compared to GW[13]and ELF. However, the weakness of G[11]and SS[24] is counterbalanced by their smaller size in terms of variables and constraints (See Table 1) which leads to a faster node solution time. Finally, it is interesting to notice that the more difficult instances are the one with density µ = 50% and percentage of negative costs ν ∈ {50%, 75%}.
3.2.2 Quadratic Knapsack Problem
Problem description. The Quadratic Knapsack Problem (QKP) consists in maximizing a quadratic function with positive coefficients subject to a linear capacity constraint. The mathematical formu-lation reads as follows:
max ( n X i=1 n X j=i+1 2Qijxixj + n X i=1 Lixi : n X i=1 cixi ≤ C, x ∈ {0, 1}n ) ,
MILP times MILP branching nodes n δ # GW[13] ELF G[11] SS[24] GW[13] ELF G[11] SS[24] 100 25 10 0.54 0.51 0.98 0.86 188.50 188.10 1004.80 717.60 50 10 4.40 4.06 2.76 2.78 438.50 405.10 2303.50 1978.10 75 10 39.39 42.23 19.96 59.65 3122.20 4212.70 9549.00 18980.50 100 10 113.82 145.09 5.87 6.73 5098.44 4918.67 7570.89 5816.89 LP times LP gaps n δ # GW[13] ELF G[11] SS[24] GW[13] ELF G[11] SS[24] 100 25 10 0.03 0.03 0.00 0.00 1.71 1.71 12.41 12.41 50 10 0.08 0.11 0.00 0.00 1.51 1.51 11.15 11.15 75 10 0.15 0.27 0.00 0.01 4.48 4.48 13.53 13.53 100 10 0.37 0.50 0.01 0.00 1.19 1.19 11.93 11.93
Table 6: Computational comparison of the different formulations for solving QKP instances
where ci (i = 1, . . . , n) are integer non negative coefficients representing the size of the items
and C is the capacity of the knapsack. In addition to the classical knapsack, where only linear profits Li are considered, here we maximize also the profits associated to couples of objects (their
profit is represented by Qij). For further information on the problem, see Caprara et al. [5] and
Pisinger [21]. In our computational tests we used randomly generated instances taken from Bil-lionnet and Soutif [3] with n = 100 and density δ = 25%, 50%, 75%, 100%, i.e., δ is the number of non-zero coefficients of the objective function divided by n(n − 1)/2.
Table description. In Table 6 we report the results concerning the computational behaviour for solving the test problems to proven optimality. Table 6 presents the same information of Table 5. Each line presents the aggregated results concerning 10 instances with same same size and δ.
Table discussion. It is interesting to notice that, when the density δ is high, the number of MILP branching nodes is comparable for all the formulations considered. Apart from this feature, the computational behaviour for the QKP of the formulations studied is in line with the one obtained for the QSSP, showing that G[11] and SS[24] are the formulations with the better computational behaviour.
4.
Conclusions
In this paper we compared several LTs for Binary Quadratic Problems present in the literature, called here GW[13], G[11]and SS[24] with a new formulation called ELF. We showed the equiva-lence of the LP-relaxation values of G[11], SS[24], GW[13]and ELF when dealing with a specific class of Unconstrained Binary Quadratic Problems. Among the formulations studied, GW[13]and ELF provide the better performances in practice when applied to unconstrained problems while G[11] and, to a lesser extent, SS[24]perform better on the special classes of constrained problems that have been tested in the paper. Whether this observation generalizes to other types of con-strained problems is an interesting question that may be worth further investigations. The new
proposed formulation ELF performs relatively well in all classes of instances. Even if GW[13]and ELF present the same LP-relaxation, their behaviour in practice can be different according to the problem considered: GW[13]is slightly preferable over ELF when dealing with Biq Mac instances, while the contrary happens when solving QSSP instances. For the two considered problems with constraints, the trade off between the strength of the LP-relaxation and the speed in the solution of the MILP branching nodes plays in favour of G[11]and SS[24]. Finally, we recall that the selection of the best LT depends also on the current performance and engineering of the MILP solvers.
5.
Acknowledgments
Thanks are due to two anonymous referees for careful reading and useful comments.
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![Table discussion. The formulations with the best computational behavior for the QSSP instances are G [ 11 ] and SS [ 24 ] on average](https://thumb-eu.123doks.com/thumbv2/123doknet/2715181.64125/19.918.139.780.108.452/table-discussion-formulations-computational-behavior-qssp-instances-average.webp)
